Bounding the smallest singular value of a random matrix without concentration

@article{Koltchinskii2013BoundingTS,
  title={Bounding the smallest singular value of a random matrix without concentration},
  author={Vladimir Koltchinskii and Shahar Mendelson},
  journal={arXiv: Probability},
  year={2013}
}
Given $X$ a random vector in ${\mathbb{R}}^n$, set $X_1,...,X_N$ to be independent copies of $X$ and let $\Gamma=\frac{1}{\sqrt{N}}\sum_{i=1}^N e_i$ be the matrix whose rows are $\frac{X_1}{\sqrt{N}},\dots, \frac{X_N}{\sqrt{N}}$. We obtain sharp probabilistic lower bounds on the smallest singular value $\lambda_{\min}(\Gamma)$ in a rather general situation, and in particular, under the assumption that $X$ is an isotropic random vector for which $\sup_{t\in S^{n-1}}{\mathbb{E}}| |^{2+\eta} \leq… 
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